Related papers: Orientation-Reversing Crystallographic Rigidity
In this paper we will develop a very general approach which shows that critical relations of holomorphic maps on the complex plane unfold transversally in a positively oriented way. We will mainly illustrate this approach to obtain…
The wavelet group and wavelet representation associated with shifts coming from a two dimensional crystal symmetry group $\Gamma$ and dilations by powers of 3, are defined and studied. The main result is an explicit decomposition of the…
We prove the rigidity of isotropic harmonic maps from a 2-torus to a complex projective space, when they are constructed from holomorphic embeddings associated to complete linear systems. We also prove that this rigidity holds for any…
This paper develops a mirror symmetry theory of Spencer cohomology within the geometric framework of constrained systems on principal bundles, revealing deep symmetric structures in constraint geometry. Based on compatible pairs…
This survey reviews results on harmonic maps into spaces of non-positive curvature, with a focus on targets that lack smooth structure. More precisely, we consider targets that are complete metric spaces with non-positive curvature in the…
We characterize the combinatorial types of symmetric frameworks in the plane that are minimally generically symmetry-forced infinitesimally rigid when the symmetry group consists of rotations and translations. Along the way, we use tropical…
We give a combinatorial characterization of generic minimally rigid reflection frameworks. The main new idea is to study a pair of direction networks on the same graph such that one admits faithful realizations and the other has only…
We prove that the linear matroid that defines generic rigidity of $d$-dimensional body-rod-bar frameworks (i.e., structures consisting of disjoint bodies and rods mutually linked by bars) can be obtained from the union of ${d+1 \choose 2}$…
Graph rigidity theory studies the capability of a graph embedded in the Euclidean space to constrain its global geometric shape via local constraints among nodes and edges, and has been widely exploited in network localization and formation…
This work investigates jammed granular matter under conditions that produce heterogeneous mass distributions on a mesoscopic scale. We consider a system of identical disks that are confined to a narrow channel, open at one end and closed…
Symmetry formulated by group theory plays an essential role with respect to the laws of nature, from fundamental particles to condensed matter systems. Here, by combining symmetry analysis and tight-binding model calculations, we elucidate…
An exactly solvable family of models describing the wrinkling of substrate-supported inextensible elastic rings under compression is identified. The resulting wrinkle profiles are shown to be related to the buckled states of an unsupported…
Tanigawa (2016) showed that vertex-redundant rigidity of a graph implies its global rigidity in arbitrary dimension. We extend this result to periodic graphs under fixed lattice representations. A periodic graph is vertex-redundantly rigid…
We study the evolution of the probability density of ensembles of iterates of the logistic map that advance towards and finally remain at attractors of representative dynamical regimes. We consider the mirror families of superstable…
A theory of free spanning sets, free bases and their space group symmetric variants is developed for the first order flex spaces of infinite bar-joint frameworks. Such spanning sets and bases are computed for a range of fundamental…
Freezing of charged colloids on square or triangular two-dimensional periodic substrates has been recently shown to realize a rich variety of orientational orders. We propose a theoretical framework to analyze the corresponding structures.…
This review describes the diversity of jammed configurations attainable by frictionless convex nonoverlapping (hard) particles in Euclidean spaces and for that purpose it stresses individual-packing geometric analysis. A fundamental feature…
Periodic space crystals are well established and widely used in physical sciences. Time crystals have been increasingly explored more recently, where time is disconnected from space. Periodic relativistic spacetime crystals on the other…
We study the rotational structures of aperiodic tilings in Euclidean space of arbitrary dimension using topological methods. Classical topological approaches to the study of aperiodic patterns have largely concentrated just on translational…
In "Universal rigidity on the line, point orde" it is shown, answering a question of Jord\'an and Nguyen, that universal rigidity of a generic bar-joint framework in R^1 depends on more than the ordering of the vertices. The graph G that…